Hyperbola
Equilateral Triangle Inscribed in Hyperbola — Area
nta_pyq_2026_jan
Grade None
Question:
Let $PQ$ be a chord of the hyperbola $\dfrac{x^2}{4}-\dfrac{y^2}{b^2}=1$, perpendicular to the $x$-axis such that $OPQ$ is an equilateral triangle, $O$ being the centre of the hyperbola. If the eccentricity of the hyperbola is $\sqrt{3}$, then the area of the triangle $OPQ$ is
Step-by-Step Solution
Key Concept: $e=\sqrt{3}$: $e^2=1+b^2/4=3\Rightarrow b^2=8$. Let $P=(x_0,y_0)$, $Q=(x_0,-y_0)$. Equilateral: $OP=PQ\Rightarrow x_0^2+y_0^2=4y_0^2\Rightarrow x_0^2=3y_0^2$.
$y_0^2=\tfrac{8}{5}$. Area $=\tfrac{8\sqrt{3}}{5}$.
Correct Answer: 4